Root locus method
- 1. Root locus method BE 5th SEM – 3rd YEAR SUB – Control System Engineering… Group no.-11 Group Members Detail 1. Vishal Kabrawala : 130090109026 2. Krunal K Patel : 130090109028 3. Kaival Patel : 130090109041 4. Ravi M Patel : 130090109045 5. Vishal Patel : 130090109046 C. K. Pithawala College Of Engineering & Technology
- 2. What is root locus……? • Simple definition : A curve or other figure formed by all the points satisfying a particular equation of the relation between coordinates, or by a point, line, or surface moving according to mathematically defined conditions. • Root locus : The root locus is the path of the roots of the characteristic equation traced out in the s-plane as a system parameter (K) is changed.(0<K<∞) • It can be used to describe qualitatively the performance of a system as various parameters are changed • It gives graphic representation of a system’s transient response and also stability • We can see the range of stability, instability, and the conditions that cause a system to break into oscillation
- 3. Root locus concept… 1+KG(s)=0 K G( s ) 1 K G( s) T.F. Characteristic equation: H(s)=1 When K=0, this collapses to D(s )= 0. Since the roots of D(s )= 0 are the poles ofG(s), those are the closed-loop poles for K=0. When K=∞, thus the roots of N(s )=0 are the zeros of G(s).
- 4. Construction of Root locus… There are many RULES which are used to construct Root locus are described below… • Rule-1 The Root locus is symmetrical about the real axis. e.g.
- 5. • Rule-2 Each branch of the root locus originates from an open loop pole at K=0 and terminates at K=∞, either on open loop zero or on infinity. The numbers of branches of the root locus terminating on infinity is equal to (n-m). Where, n= numbers of poles m=numbers of zeros e.g.
- 6. • Rule-3 Segments of the real axis having an odd number of real axis open loop poles plus open loop zeros to their right are parts of the root locus. If numbers of (poles + zeros) at right side to the segment = ODD, then it is a part of Root locus, else it is not. e.g.
- 7. • Rule-4 (n-m) is equals to numbers of asymptotes For to find asymptotes making angles with real axis… X=0,1,…(n-m-1) β =180(2X+1)/(n-m) e.g.
- 8. • Rule-5 The point of intersection of the asymptotes with the real axis is at e.g. mn zp n i m j ji a 1 1
- 9. • Rule-6 The breakaway point of the root locus are determined from the roots of the equation S should be real for breakaway point. If it is not real, then no breakaway point. e.g. 0 dS dK
- 10. • Rule-7 The intersection of the root locus branches with the imaginary axis can be determined by use of the routh criterion. e.g.
- 11. Rule 8 is only applicable when either complex poles or complex zeros are there • Rule-8 The angle of departure suppose… )22)(22( jSjSS K GH 22 )22( js jSS K GH )2222)(22( jjj K GH 00 0 90135 0 ' AngGH 0 225' AngGH '180 AngGHD jsatD 22_45 jsatD 22_45
- 13. Ex. 1.
- 18. Ex. 2.